IB Mathematics: AA HL topic guide

Functions

Functions is a core part of IB Mathematics: AA HL. This guide connects the syllabus ideas behind Introduction to Functions, The Concept of a Function, Composite and Inverse Functions, Transformations of Graphs, Quadratic Functions and 4 more units, shows how they appear in worked problems, and points you to the formulas and full lessons needed for exam revision.

What you will learn

Functions syllabus outline

The units below follow the structure used in the full Study to Learn course. Use the outline to identify exactly which idea needs attention, then work through the public example before continuing to the complete lesson path.

2.0

Introduction to Functions

What You Will Learn in Functions

2.1

The Concept of a Function

Domain, Range & Function Notation · Key Features of Graphs

2.2

Composite and Inverse Functions

Composite Functions · Inverse Functions

2.3

Transformations of Graphs

Translations & Reflections · Stretches & Composite Transformations

2.4

Quadratic Functions

Forms of a Quadratic & the Discriminant · Applications of Quadratics

2.5

Rational, Exponential and Logarithmic Functions

The Reciprocal & Rational Functions · Exponential & Logarithmic Graphs

2.6

Solving Equations and Modelling with Functions

Solving Equations Graphically & Analytically · Modelling Real-World Data

2.7

Polynomial Functions

The Factor & Remainder Theorems · Further Polynomial Analysis

2.8

Further Rational and Special Functions

Rational Functions with Oblique Asymptotes · Odd and Even Functions

Free worked preview

The Factor & Remainder Theorems

This complete preview comes from the Polynomial Functions unit. It introduces the core language, shows the method in context, and gives you a real example of the lesson quality before you create an account.

The Remainder Theorem

Factoring quadratics is straightforward; factoring cubics and higher-degree polynomials is not. The remainder and factor theorems give you the systematic entry point: test candidate roots (using the rational root theorem), and when

P(a)=0

, you know

(x-a)

is a factor. This HL-only topic extends your algebraic toolkit to higher-degree polynomials and is essential preparation for sketching polynomial curves and for the polynomial content in the calculus sections. By the end you will be able to use the remainder theorem to find remainders without long division, apply the factor theorem to find factors of polynomials, and fully factorise cubic and quartic polynomials.

Remainder Theorem

\text{If } P(x) \text{ is divided by } (x-a),\text{ the remainder is } P(a)

This lets you find a division remainder instantly, without performing the division.

The Factor Theorem

Factor Theorem

(x-a) \text{ is a factor of } P(x) \iff P(a) = 0

This is the remainder theorem's special case when the remainder is zero — it's the standard tool for factoring cubics and higher-degree polynomials once one root is found (often by testing small integer factors of the constant term).

Common ErrorWhen testing candidate roots, only factors of the constant term (divided by factors of the leading coefficient — the Rational Root Theorem) are worth testing systematically; testing random values wastes time.

Worked Example Factor

P(x) = x^3 - 2x^2 - 5x + 6

completely.

1Test x=1 (a factor of the constant term 6) — it works, so (x−1) is a factor

P(1) = 1-2-5+6 = 0

2Divide P(x) by (x−1), e.g. by polynomial long division

P(x) = (x-1)(x^2-x-6)

3Factor the remaining quadratic

x^2-x-6 = (x-3)(x+2)

P(x) = (x-1)(x-3)(x+2)

Practice QuestionUse the factor theorem to show that (x+2) is a factor of

x^3+3x^2-4x-12

, then fully factor it.

Reviewed by the Study to Learn editorial team · Updated 2026-07-24